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Metamath Proof Explorer

Theorem dochssv

Description: A subspace orthocomplement belongs to the DVecH vector space. (Contributed by NM, 22-Jul-2014)

Ref Expression
Hypotheses dochssv.h 
|- H = ( LHyp ` K )
dochssv.u 
|- U = ( ( DVecH ` K ) ` W )
dochssv.v 
|- V = ( Base ` U )
dochssv.o 
|- ._|_ = ( ( ocH ` K ) ` W )
Assertion dochssv 
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( ._|_ ` X ) C_ V )

Proof

Step Hyp Ref Expression
1 dochssv.h 
 |-  H = ( LHyp ` K )
2 dochssv.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 dochssv.v 
 |-  V = ( Base ` U )
4 dochssv.o 
 |-  ._|_ = ( ( ocH ` K ) ` W )
5 eqid 
 |-  ( ( DIsoH ` K ) ` W ) = ( ( DIsoH ` K ) ` W )
6 1 5 2 3 4 dochcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( ._|_ ` X ) e. ran ( ( DIsoH ` K ) ` W ) )
7 1 2 5 3 dihrnss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ._|_ ` X ) e. ran ( ( DIsoH ` K ) ` W ) ) -> ( ._|_ ` X ) C_ V )
8 6 7 syldan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( ._|_ ` X ) C_ V )